1. Field of Invention
The invention relates to a method for estimating the life of an industrial apparatus using gas, or the like. More particularly, the invention is concerned with a method of estimating the life of a gas-using apparatus or the like by treating a damage cumulating process of each component of the apparatus as a stochastic process.
2. Description of Related Art
For gas apparatus materials for high temperatures, including industrial furnaces, there is no common standard as to when and how inspection is to be conducted, and measures are taken according to the purposes for which the apparatuses are used. In many cases, gas apparatuses are used in environments which are severe thermally and chemically, such as environments exposed to high temperatures or apt to undergo corrosion. Even in the case of apparatuses having the same specifications, loads imposed thereon differ depending on users and there occur relatively large variations in the accumulation speed of apparatus damage or in the apparatus life. Monitoring the state of apparatus components in detail may be a way to solve this problem, but there arise such problems as the sensor operation environment and the place of installation being limited and the cost for the monitor causing a cost increase. Thus, at present, there are few techniques for practical application.
Particularly, in a gas apparatus under working conditions, starting and stopping of operation are repeated in accordance with an operation schedule of the apparatus and there occur variations in the amount of heat transferred to an article to be heated, for example, and a narrow-band random stress amplitude variation involving a relatively random variation in peak values of a load stress, such as a thermal stress, is applied to the material of the apparatus. The xe2x80x9cnarrow bandxe2x80x9d means that variations in the peak value of a load stress, such as a thermal stress, are in a relatively narrow range.
Moreover, in a high-temperature gas apparatus it is presumed that there will occur damage caused by creep deformation. Creep deformation indicates a deformation caused by an increase of strain with the lapse of time upon exertion of a certain magnitude of stress on a certain material under a half, or higher, temperature of a melting point at absolute temperature.
For this reason, in the development of a high-temperature gas apparatus, it is considered necessary to develop a damage estimating technique capable of estimating damage accumulation caused by load variations under working conditions.
One such known damage estimating technique is a technique in which a material damage process is treated as a stochastic process. In connection with this technique, the following two methods are known.
In the first method, the development of a crack in a material is treated as a stochastic process. Further, in connection with causes of irregularity in a damage development model, classification can be made into studies in which a crack development resistance is adopted and studies in which irregularity of load stresses is adopted.
In these studies, basically a random term which is a source of irregularity is introduced in part of Paris-Erdogan""s law, which is a deterministic equation representing crack development, independently of the cause of irregularity, to afford a stochastic differential equation, thereby building a model of damage development.
In the second method, which is based on the concept of continuum damage dynamics, the influence of a fluctuating load and a time and spatial variation in a microscopic material characteristic caused by the occurrence of a microcrack, or the like, upon a change in a macroscopic characteristic of the material strength is formulated and the development of damage is described. This method is one of the practical methods because it handles a damage parameter which can be defined from a macroscopic characteristic.
As a typical example of the above method there is known a study made by Silberschmidt. In this study, a non-linear Langevin equation (expression 1) is given for damage accumulation of a randomly fluctuating minor-axis tensile load (I mode):                                                         ⅆ              p                                      ⅆ              t                                =                                    f              ⁡                              (                p                )                                      +                                          g                ⁡                                  (                  p                  )                                            ⁢                              xe2x80x83                            ⁢                              L                ⁡                                  (                  t                  )                                                                    ,                            (        1        )            
where f(p) is the right side of a deterministic equation for mode I damage:
f(p)=Ap3+Bp2+Cpxe2x88x92D"sgr",xe2x80x83xe2x80x83(2)
and L(t) is a stochastic term, A, B, C, and D are empirical values, and g(p) is modeled on the assumption that the strength of the stochastic term is proportional to the accumulation degree of damage at a certain time. In the Silberschmidt""s analysis, the non-linear Langevin equation is solved numerically to indicate a qualitative change of PDF (probability density function) against a change in stress variation strength, and an empirical fact for the shortening of the material life, which occurs in the presence of stress variation, is shown by calculation.
However, the conventional methods for estimating the life of a gas apparatus involve the following problems.
In the above first method, because the calculation is made on the basis of the development of crack, it is necessary to determine which portion of the apparatus is apt to crack. Generally, a crack-prone place is determined on the basis of a portion of the apparatus where stress concentration is apt to occur. But the components of the gas apparatus operating in a production site are complicated in shape, so it is in many cases difficult to predict a portion of the apparatus where a crack is apt to occur. Also due to the complicated shapes of the gas apparatus components, the process up to rupture may differ greatly depending on the crack-formed places.
Upon occurrence of a crack it is necessary to check the state of the crack in detail, which, however, is difficult because of complicated shapes of gas apparatus components.
Therefore, in estimating with a high accuracy the life of a gas apparatus working in a production site, it is in many cases difficult to adopt a method which involves making a direct calculation for a crack while regarding the crack as being clear in its size and position, thereby introducing a random term as a source of irregularity into part of the Paris-Erdogan""s law which is a deterministic equation representing basically the development of the crack, to afford a stochastic differential equation, and thereby building a model of damage development.
In connection with the above second method, the method of estimating the creep life of a gas apparatus is advantageous in that it is not necessary to take the development of a crack into account. But no reference is made therein to temperature variation and it is impossible to estimate the influence of temperature variation. When there is a temperature variation, therefore, it is impossible to accurately estimate the creep life. In gas apparatuses, however, not only stress but also temperature varies in many cases, in which case the method in question is not applicable.
Thus, it is difficult for the second method to accurately estimate the life of a gas apparatus.
The invention has been accomplished to solve the above-mentioned problems and it is an object of the invention to provide a method wherein, when treating a damage process of material as a stochastic process, the life of an apparatus under a narrow-band random stress variation is estimated without making a direct calculation while regarding a crack as being clear in its size and position.
It is also an object of the invention to provide a method wherein, when treating a damage process of material as a stochastic process, the influence of a fluctuating load and a time and spatial variation in a microscopic material characteristic caused by the occurrence of a microcrack or the like upon a change in a macroscopic characteristic of the material strength is formulated. The development of damage is then described to estimate a creep life of the apparatus concerned, the creep life estimation being done in the case where both narrow-band random stress variation and narrow-band random temperature variation are applied to the apparatus.
To achieve the above-mentioned objects of the invention, there is provided a method for estimating a life of an apparatus under a random stress amplitude variation, involving determining a probability density function of a cumulated damage quantity and estimating the life of the apparatus on the basis of the probability density function, characterized by approximating a damage coefficient indicative of a damage quantity per unit by a linear expression when the random stress amplitude variation is in a narrow band; and representing the random stress amplitude variation "sgr"(t)(instantaneous) in terms of the sum of a time averaged value "sgr"(t)(mean) and a stochastic variation "sgr"xe2x80x2.
In the apparatus life estimating method under a narrow-band random stress variation, which has the above-mentioned characteristics, Miner""s law is used. By Miner""s law is meant a method wherein an accumulated damage quantity is calculated by accumulating a life which is determined by both stress and repetitive number with use of an S-N curve, and a residual life is estimated. Thus, it is not necessary to use Paris-Erdogan""s law, which is a deterministic equation representing the development of a crack, that is, no consideration is needed of the development of a crack. Further, by representing the random stress amplitude variation "sgr"(t)(instantaneous) in terms of the sum of both time averaged value "sgr"(t)(mean) and stochastic variation "sgr"xe2x80x2(t) and by approximating a damage coefficient by a linear expression which coefficient represents a damage quantity for one time, there is derived a Langevin equation of the accumulated damage quantity which represents Miner""s law. The Langevin equation of the cumulated damage quantity which represents Miner""s law indicates a stochastic differential equation with a stochastic process-containing function introduced into a dynamic equation which represents the development of damage shown by Miner""s law in the case of the stress amplitude being constant. Consequently, Miner""s law is extended in the case where the load stress amplitude varies randomly in a narrow band.
Thus, a model of the development of accumulated damage quantity can be shown by solving this Langevin equation and therefore a mean value or a deviation of damage accumulated in a material at a certain time can be obtained without directly handling a crack which is clear in its size and position.
The invention is also characterized by using as the above damage accumulation process a Langevin equation and a Fokker-Planck equation corresponding thereto.
That is, in estimating material damage and life, not only a mean value and a deviation of the damage accumulated in the material at a certain time, but also a probability density function and a probability distribution of damage play an important role. Generally, the probability density function of damage is arranged in terms of a normal distribution, a logarithmic normal distribution, or a Weibull distribution. But a distribution in the case of a randomly fluctuating stress amplitude is not clear at present. Therefore, a Fokker-Planck equation corresponding to the Langevin equation is derived. The Fokker-Planck equation indicates a partial differential equation of second order in a probability density function derived on the assumption that a moment of a cubic or higher order of the transition quantity can be ignored in a continuous Markov process. The Markov process indicates a process in which information at a future time t2 relating to a stochastic variable is described completely by information at the present time t1.
Accordingly, by solving the Fokker-Planck equation, a probability density function of a cumulated damage quantity at any time in the period from the start of the experiment up to rupture can be expressed in the form of a normal distribution.
Further, on the basis of the Fokker-Planck equation, it is possible to obtain a predictive expression of a residual life from an arbitrary cumulated damage quantity of a material which has already been damaged. Thus, even in the case of a randomly varying stress amplitude, it is possible to obtain a probability density function of damage and a predictive expression of a residual life.
In the creep life estimating method according to the invention, a damage coefficient based on Robinson""s damage fraction rule is used to determine a probability density function of a cumulated damage quantity. According to the method using Robinson""s damage fraction rule, an accumulated damage quantity is calculated by accumulating a life determined by a degree-of-damage curve which uses the Larson-Miller parameter plotted along the axis of abscissa and stress plotted along the axis of ordinate. The Larson-Miller parameter is an empirical function with stress being represented by both temperature and life in creep rupture. Thus, both stress and temperature can be taken into consideration in the estimation of life.
Moreover, by representing the random stress amplitude variation "sgr"(t)(instantaneous) in terms of the sum of time averaged value "sgr"(t)(mean) and stochastic variation "sgr"xe2x80x2(t), by representing the random temperature variation xcex8(t)(instantaneous) in terms of the sum of time averaged value xcex8(t)(mean) and stochastic variation xcex8xe2x80x2(t), and further by approximating the damage coefficient which represents the damage quantity for one time by a linear expression, there is derived a Langevin equation of an accumulated damage quantity. The Langevin equation of an accumulated damage quantity means a stochastic differential equation with a function incorporated in a dynamic equation which represents a damage evolution shown by the Robinson""s damage fraction rule in a constant temperature condition, the function containing a stochastic process based on stress variation and temperature variation. With the stochastic differential equation, the Robinson""s damage fraction rule is extended in the case where both load stress and load temperature vary in a narrow band.
By solving the Langevin equation it is possible to show a development model of the accumulated damage quantity based on creep deformation in case of both load stress and load temperature varying randomly in a narrow band. That is, it is possible to accurately estimate the life of a gas apparatus in which both stress and temperature fluctuate.
The invention is further characterized by using, as the damage cumulation process, both a Langevin equation and a Fokker-Planck equation corresponding thereto.
That is, a Fokker-Planck equation corresponding to the Langevin equation is derived. The Fokker-Planck equation means a partial differential equation of second order in a probability density function which has been derived on the assumption that a moment of cubic or higher order of the transition quantity can be ignored in a continuous Markov process. The Markov process indicates a process wherein information at a future time t2 relating to a stochastic variable is described completely by information at the present time t1.
By solving the Fokker-Planck equation, a probability density function of a cumulated damage quantity at any time in the period from the start of the experiment up to rupture can be expressed in the form of a normal distribution.
Further, on the basis of the Fokker-Planck equation it is possible to obtain a predictive expression of a residual life from an arbitrary cumulated damage quantity of a material which has already been damaged. Thus, it is possible to obtain a probability density function of damage and a predictive expression of a residual life in the case where both stress and temperature vary randomly.